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In mathematics, the Lehmer mean of a tuple of positive real numbers, named after Derrick Henry Lehmer,[1] is defined as:
The weighted Lehmer mean with respect to a tuple of positive weights is defined as:
The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.
Properties
The derivative of is non-negative
thus this function is monotonic and the inequality
holds.
The derivative of the weighted Lehmer mean is:
Special cases
- is the minimum of the elements of .
- is the harmonic mean.
- is the geometric mean of the two values and .
- is the arithmetic mean.
- is the contraharmonic mean.
- is the maximum of the elements of . Sketch of a proof: Without loss of generality let be the values which equal the maximum. Then
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